PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 9, September 2008, Pages 3185–3193 S 0002-9939(08)09299-X Article electronically published on April 29, 2008 SELF-COMMUTATORS OF AUTOMORPHIC COMPOSITION OPERATORS ON THE DIRICHLET SPACE A. ABDOLLAHI (Communicated by N. Tomczak-Jaegermann) Abstract. Let ϕ be a conformal automorphism on the unit disk U and C ϕ : D-→D be the composition operator on the Dirichlet space D in- duced by ϕ. In this article we completely determine the point spectrum, spec- trum, essential spectrum and essential norm of the operators C ∗ ϕ C ϕ ,C ϕ C ∗ ϕ and self-commutators of C ϕ , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators. 1. Introduction Let ϕ be a holomorphic self-map of the unit disk U := {z ∈ C : |z| < 1}. The function ϕ induces the composition operator C ϕ , defined on the space of holomor- phic functions on U by C ϕ f = f ◦ ϕ. The restriction of C ϕ to various Banach spaces of holomorphic functions on U has been an active subject of research for more than three decades, and it will continue to be for decades to come (see [15], [16] and [5]). Let D denote the Dirichlet space of analytic functions on the unit disk with derivatives that are square integrable with respect to the area measure on the disk. In recent years the study of composition operators on the the Dirichlet space has received considerable attention (see [6], [3], [7], [10], [11] and the references cited therein). The main aim here is to find the spectrum, point spectrum, essential spectrum and essential norm of C ∗ ϕ C ϕ ,C ϕ C ∗ ϕ , self-commutator [C ∗ ϕ ,C ϕ ]= C ∗ ϕ C ϕ − C ϕ C ∗ ϕ and anti-self-commutator {C ∗ ϕ ,C ϕ } = C ∗ ϕ C ϕ + C ϕ C ∗ ϕ , for automorphic composition operators C ϕ on the Dirichlet space. Actually, this work is an extension of the work of an earlier paper by P. S. Bour- don and B. MacCluer [2] to the Dirichlet space. In [2], by using Cowen’s formula for the adjoint of C ϕ on H 2 (U), the authors have completely determined the spectrum, essential spectrum and point spectrum for self-commutators of automorphic compo- sition operators acting on the Hardy space of unit disk. By using Cowen’s formula extensions to A 2 (B N ) and H 2 (B N ) for N ≥ 1, they have also shown that, for each Received by the editors May 14, 2007, and, in revised form, July 16, 2007. 2000 Mathematics Subject Classification. Primary 47B33; Secondary 47A10, 47E20, 47B47. Key words and phrases. Dirichlet space, composition operator, spectrum, essential spectrum, essential norm, self-commutator, anti-self-commutator. This research was partially supported by a grant from the Shiraz University Research Council. This work was carried out at the Department of Mathematics, University of Auckland, where the author was on the sabbatical leave during the academic year 2006-2007. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 3185 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use